New predictions for $\Lambda_b\to\Lambda_c$ semileptonic decays and tests of heavy quark symmetry

The heavy quark effective theory makes model independent predictions for semileptonic $\Lambda_b \to \Lambda_c$ decays in terms of a small set of parameters. No subleading Isgur-Wise function occurs at order $\Lambda_{\rm QCD}/m_{c,b}$, and only two sub-subleading functions enter at order $\Lambda_{\rm QCD}^2/m_c^2$. These features allow us to fit the form factors and decay rates calculated up to order $\Lambda_{\rm QCD}^2/m_c^2$ to LHCb data and lattice QCD calculations. We derive a significantly more precise standard model prediction for the ratio ${\cal B}(\Lambda_b\to \Lambda_c \tau\bar\nu) / {\cal B}(\Lambda_b\to \Lambda_c \mu\bar\nu)$ than prior results, and find the expansion in $\Lambda_{\rm QCD}/m_c$ well-behaved, addressing a long-standing question. Our results allow more precise and reliable calculations of $\Lambda_b\to \Lambda_c\ell\bar\nu$ rates, and are systematically improvable with better data on the $\mu$ (or $e$) modes.

where l = µ, e. Combining the D and D * results, the tension with the SM is 4σ [1]. Precision control of hadronic matrix elements are crucial to predict the ratios of decay rates; A better understanding of the heavy quark expansion to O(Λ 2 QCD /m 2 c ) is required, as it is largely responsible for the different uncertainty estimates of R(D * ) in the SM [2][3][4]. The same hadronic matrix elements are also crucial to resolve tensions between inclusive and exclusive determinations of |V cb | [2][3][4][5][6][7][8][9].
The baryonic Λ b → Λ c ν decays provide a theoretically cleaner laboratory than B → D ( * ) ν to examine O(Λ 2 QCD /m 2 c ) terms, as heavy quark symmetry [10][11][12] provides stronger constraints. The O(Λ QCD /m c,b ) contributions yield no new nonperturbative functions beyond the leading order Isgur-Wise function, significantly reducing the number of hadronic parameters order by order in the heavy quark effective theory (HQET) [13,14] description of these decays. This allows us to determine to the O(Λ 2 QCD /m 2 c ) contributions to an exclusive decay for the first time, without any model dependent assumptions.
In this letter we examine the HQET predictions at O(Λ 2 QCD /m 2 c ) and fit them to a recent LHCb measurement of Λ b → Λ c µν [15] and/or lattice QCD (LQCD) results [16]. Doing so, we obtain the most precise SM prediction so far for improvable with future data. We find that the O(Λ 2 QCD /m 2 c ) corrections have the expected characteristic size, suggesting that the heavy quark expansion is well behaved in such decays at the m c scale.
Testing HQET predictions not only provides a path to reducing theoretical uncertainties in precision R(D ( * ) ) predictions and extraction of |V cb |, but also improves the sensitivity to possible new physics contributions. Measuring semileptonic decays mediated by the same partonlevel transition between different hadrons is important, as they improve the statistics, entail different systematic uncertainties, and give complementary information on possible new physics. LHCb projections show that the precision of R(Λ c ) will be near those of R(D ( * ) ) in the future [17], making this channel very important.

HQET EXPANSION OF THE FORM FACTORS
The semileptonic Λ b → Λ c ν form factors in HQET are conventionally defined for the SM currents as [18][19][20] , and the f i and g i form factors are functions of w = v · v . The spinors are normalized toūu = 2m.
The Λ b,c baryons are singlets of heavy quark spin symmetry, with the "brown muck" of the light degrees of freedom in the spin-0 ground state. Therefore, where Q = b, c, the ellipsis denotes terms higher order in Λ QCD /m Q and m Λ b = 5.620 GeV, m Λc = 2.286 GeV [21]. The parameterΛ Λ is the energy of the light degrees of freedom in the m Q Λ QCD limit, and λ Λ 1 is related to the heavy quark kinetic energy in the Λ b,c baryons. As in Ref. [22], we use the 1S scheme [23][24][25] to eliminate the leading renormalon ambiguities in the quark masses (and arXiv:1808.09464v1 [hep-ph] 28 Aug 2018 hence inΛ Λ ). That is, we treat m 1S b = (4.71 ± 0.05) GeV and δm bc = m b − m c = (3.40 ± 0.02) GeV as independent parameters [26]. (The latter is well constrained by B → X c ν spectra [27,28].) We match HQET onto QCD at scale µ = √ m c m b , so that α s 0.26. For example, using Eq. (4) for both Λ b and Λ c to eliminate λ Λ 1 , at O(α s ) we obtainΛ Λ = (0.81 ± 0.05) GeV.
At order Λ QCD /m c,b a remarkable simplification occurs compared to meson decays: The O(Λ QCD /m c,b ) corrections from the matching of thec Γb heavy quark current onto HQET [29][30][31] can be expressed in terms ofΛ Λ and the leading order Isgur-Wise function ζ(w) [32]. In addi- contributions from the chromomagnetic operator. The kinetic energy operator in the O(Λ QCD /m c,b ) HQET La-grangian gives rise to a heavy quark spin symmetry conserving subleading term, parametrized by ζ ke (w), which can be absorbed into the leading order Isgur-Wise function by redefining ζ as where ε c,b =Λ Λ /(2 m c,b ). Thus, no unknown functions of w are needed to parametrize the O(Λ QCD /m c,b ) corrections. Luke's theorem [33] implies ζ ke (1) = 0, so the normalization ζ(1) = 1 is preserved. Perturbative corrections to the heavy quark currents can be computed by matching QCD onto HQET [29][30][31], and introduce no new hadronic parameters.

FITS TO LHCb AND LATTICE QCD DATA
To determine the nonperturbative quantities that occur in the HQET expansion of the form factors in Eq. (8), assess the behavior of the expansion in Λ QCD /m c , and derive precise SM predictions for R(Λ c ) in Eq. (2), we fit the LHCb measurement of dΓ(Λ b → Λ c µν)/dq 2 [15] or/and a LQCD determination of the six form factors [16].
The LHCb experiment measured the q 2 spectrum in 7 bins, normalized to unity [15]. This reduces its effective degrees of freedom from 7 to 6 (as any one bin is determined by the sum of the others). The measurement is shown as the data points in Fig. 1.
The lattice QCD results [16] for the 6 form factors are published as fits to the BCL parametrization [40], using either 11 or 17 parameters. We derive predictions for f 1,2,3 and g 1,2,3 using the 17 parameter result at three q 2 values, q 2 = 1 GeV 2 , q 2 max /2, q 2 max − 1 GeV 2 , preserving their full correlation structure in order to construct an appropriate covariance matrix. The difference in the form factor values obtained using the 17 or the 11 parameter results is added as an additional uncorrelated uncertainty. This slightly differs from the prescription proposed in Ref. [16], using the maximal differences, which cannot preserve the correlation structure between the form factor values. The 18 form factor values used in our fits are shown as data points in Fig. 2. The LQCD predictions, following the prescription of Ref. [16], are shown as heather gray bands, and the uncertainties are in good agreement. The heather gray band in Fig. 1 shows the LQCD prediction for the normalized spectrum, using the BCL parametrization.
The SM prediction for the decay rate for arbitrary charged lepton mass is Combined with f 1 = f ⊥ and g 1 = g ⊥ , Eqs. (10) relate f i and g i to the other common form factor basis, f ⊥,+,0 and g ⊥,+,0 , used in Ref. [16]. Our result in Eq. (9) agrees with those in Refs. [16,41].
In our fits to the LHCb data, we integrate the rate predictions that follow from Eqs. (8) and (9) 1. The red band shows our fit of the HQET predictions to dΓ(Λ b → Λcµν)/dq 2 measured by LHCb [15] and the LQCD form factors [16]. The heather gray band shows the LQCD prediction. The blue curve shows our prediction for dΓ(Λ b → Λcτν)/dq 2 .
bin, and minimize a χ 2 function. The LQCD predictions are fitted by minimizing a χ 2 function that includes the 18 values and their correlations, as described above.
We explore three scenarios: (i) fitting only the LHCb spectrum; (ii) fitting only the LQCD data; and (iii) a combined fit the the LHCb data and the LQCD information. The resulting HQET parameters are summarized in Table I. For the fit to only the LHCb spectrum, the unknown absolute normalization of the measurement removes the sensitivity tob 1,2 . Therefore, we constrain them to 0 by a Gaussian with a 2 GeV 2 (≈ 3Λ 2 Λ ) uncertainty, motivated by a model dependent estimate for b 1 (1) [19]. This allows our 3 fits to have the same parameters, and be compared to one another. In all fits, m 1S b and δm bc are constrained using Gaussian uncertainties. The leading order Isgur-Wise function is fitted as   8) to the LQCD results [16] and the LHCb spectrum [15] for the 6 form factors (red bands). The heather gray bands and data points show the LQCD prediction, see text for details.
using instead of w the conformal parameters z or z * yield nearly identical fits. Fits with ζ linear in either w, z, or z * are poor. Adding more q 2 values from the BCL fit of the LQCD result to our sampling indicates no preference for the inclusion of higher order terms in w − 1. We fitb 1,2 as constants, which is appropriate at the current level of sensitivity. We do not include explicitly an uncertainty for neglected higher order terms in Eq. (8); two form factors, f 3 and g 3 , receive no Λ 2 QCD /m 2 c corrections, so their agreement with LQCD in the right-most plots in Fig. 2 indicates that those are probably small.
All fits have acceptable χ 2 values, and they all yield compatible values for the slope and curvature of the leading Isgur-Wise function. The fit of the HQET predictions to the lattice QCD form factors determines fairly precisely theb 1,2 parameters, which occur at O(Λ 2 QCD /m 2 c ). The significance ofb 1 = 0 is over 3σ. Howeverb 1 (1) is much smaller than the model dependent estimatê b 1 (1) −3Λ 2 Λ , obtained in Eq. (5.5) of Ref. [19]. The red bands in Figures 1 and 2 show the combined fit using both LHCb and LQCD information. This agreement shows that the HQET predictions in Eq. (8 describe the form factors and the experimental spectrum at the current level of uncertainties. This also holds for the fit using the LHCb spectrum (and constraints onb 1,2 ). Table II shows the correlation matrix of the LHCb + LQCD fit. Table I also shows our SM predictions for R(Λ c ) from each of the 3 fits, and Fig. 1 shows the predicted differential rate dΓ(Λ b → Λ c τν)/dq 2 as a blue band.

CONCLUSIONS
Measurements of Λ b → Λ c ν decays will play important roles in elucidating the tantalizing hints of new physics in the measurements of R(D ( * ) ), and refining our understanding of determinations of the CKM element |V cb |. We derived new model independent predictions for these decays, and found that fitting the LHCb data for dΓ(Λ b → Λ c µν)/dq 2 substantially reduces the uncertainty of the SM prediction for R(Λ c ). We obtained R(Λ c ) = 0.324 ± 0.004 .
Combining the lattice information with the measured spectrum produces the most precise prediction of R(Λ c ) to date, significantly improving the precision over the lattice QCD prediction, R(Λ c ) = 0.3328 ± 0.0070 ± 0.0074 [16]. This large improvement arises because the experimental data constrains combinations of form factors relevant for the prediction of R(Λ c ). Using the lattice QCD form factor calculations, we perform new tests of heavy quark symmetry, determining Λ 2 QCD /m 2 c corrections to an exclusive decay, without any model dependent assumptions, for the first time. We find good agreement between lattice QCD and HQET predictions. In particular, the HQET expansion at order Λ 2 QCD /m 2 c appears well behaved. More details and extensions of these results including new physics contributions will be presented elsewhere [42].
We thank the Aspen Center of Physics, supported by the NSF grant PHY-1066293, where parts of this work were completed. We thank HG for inspiring color choices to scoop FT. FB thanks JMB for seven wonderful years together. FB and WS were supported by the DFG Emmy-Noether Grant No. BE 6075/1-1. ZL was supported in part by the U.S. Department of Energy under contract DE-AC02-05CH11231. The work of DR was supported in part by NSF grant PHY-1720252.